1,671 research outputs found
The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues
Let G be a distance-regular graph with diameter d and Kneser graph K=Gd, the distance-d graph of G. We say that G is partially antipodal when K has fewer distinct eigenvalues than G. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with d distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.Peer ReviewedPostprint (author's final draft
The spectral excess theorem for graphs with few eigenvalues whose distance- 2 or distance-1-or-2 graph is strongly regular
We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Î (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d+1 is the number of different eigenvalues of Î. This can be seen as another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.Research of C. DalfĂł and M. A. Fiol is partially supported by AgĂšncia de GestiĂł d'Ajuts Universitaris i de Recerca (AGAUR) under project 2017SGR1087. Research of J. Koolen is partially supported by the National Natural Science Foundation of China under project No. 11471009, and the Chinese Academy of Sciences under its â100 talentâ programme. The research of C. DalfĂł has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie SkĆodowska-Curie grant agreement No. 734922
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Explicit near-Ramanujan graphs of every degree
For every constant and , we give a deterministic
-time algorithm that outputs a -regular graph on
vertices that is -near-Ramanujan; i.e., its eigenvalues
are bounded in magnitude by (excluding the single
trivial eigenvalue of~).Comment: 26 page
Detection thresholds in very sparse matrix completion
Let be a rectangular matrix of size and be the random
matrix where each entry of is multiplied by an independent
-Bernoulli random variable with parameter . This paper is about
when, how and why the non-Hermitian eigen-spectra of the randomly induced
asymmetric matrices and captures more of the
relevant information about the principal component structure of than via
its SVD or the eigen-spectra of and , respectively. Hint: the
asymmetry inducing randomness breaks the echo-chamber effect that cripples the
SVD.
We illustrate the application of this striking phenomenon on the low-rank
matrix completion problem for the setting where each entry is observed with
probability , including the very sparse regime where is of order ,
where matrix completion via the SVD of fails or produces unreliable
recovery. We determine an asymptotically exact, matrix-dependent, non-universal
detection threshold above which reliable, statistically optimal matrix recovery
using a new, universal data-driven matrix-completion algorithm is possible.
Averaging the left and right eigenvectors provably improves the recovered
matrix but not the detection threshold. We define another variant of this
asymmetric procedure that bypasses the randomization step and has a detection
threshold that is smaller by a constant factor but with a computational cost
that is larger by a polynomial factor of the number of observed entries. Both
detection thresholds shatter the seeming barrier due to the well-known
information theoretical limit for matrix completion found in
the literature.Comment: 84 pages, 10 pictures. Submitte
Improvements on non-equilibrium and transport Green function techniques: the next-generation transiesta
We present novel methods implemented within the non-equilibrium Green
function code (NEGF) transiesta based on density functional theory (DFT). Our
flexible, next-generation DFT-NEGF code handles devices with one or multiple
electrodes () with individual chemical potentials and electronic
temperatures. We describe its novel methods for electrostatic gating, contour
opti- mizations, and assertion of charge conservation, as well as the newly
implemented algorithms for optimized and scalable matrix inversion,
performance-critical pivoting, and hybrid parallellization. Additionally, a
generic NEGF post-processing code (tbtrans/phtrans) for electron and phonon
transport is presented with several novelties such as Hamiltonian
interpolations, electrode capability, bond-currents, generalized
interface for user-defined tight-binding transport, transmission projection
using eigenstates of a projected Hamiltonian, and fast inversion algorithms for
large-scale simulations easily exceeding atoms on workstation computers.
The new features of both codes are demonstrated and bench-marked for relevant
test systems.Comment: 24 pages, 19 figure
Topological characteristics of IP networks
Topological analysis of the Internet is needed for developments on network planning, optimal routing
algorithms, failure detection measures, and understanding business models. Accurate measurement, inference and modelling techniques are fundamental to Internet topology research. A requirement towards
achieving such goals is the measurements of network topologies at different levels of granularity. In this
work, I start by studying techniques for inferring, modelling, and generating Internet topologies at both
the router and administrative levels. I also compare the mathematical models that are used to characterise
various topologies and the generation tools based on them.
Many topological models have been proposed to generate Internet Autonomous System(AS) topologies. I use an extensive set of measures and innovative methodologies to compare AS topology generation models with several observed AS topologies. This analysis shows that the existing AS topology
generation models fail to capture important characteristics, such as the complexity of the local interconnection structure between ASes. Furthermore, I use routing data from multiple vantage points to show
that using additional measurement points significantly affect our observations about local structural properties, such as clustering and node centrality. Degree-based properties, however, are not notably affected
by additional measurements locations. The shortcomings of AS topology generation models stems from
an underestimation of the complexity of the connectivity in the Internet and biases of measurement techniques.
An increasing number of synthetic topology generators are available, each claiming to produce
representative Internet topologies. Every generator has its own parameters, allowing the user to generate
topologies with different characteristics. However, there exist no clear guidelines on tuning the value of
these parameters in order to obtain a topology with specific characteristics. I propose a method which
allows optimal parameters of a model to be estimated for a given target topology. The optimisation
is performed using the weighted spectral distribution metric, which simultaneously takes into account
many the properties of a graph.
In order to understand the dynamics of the Internet, I study the evolution of the AS topology over a
period of seven years. To understand the structural changes in the topology, I use the weighted spectral
distribution as this metric reveals differences in the hierarchical structure of two graphs. The results indicate that the Internet is changing from a strongly customer-provider oriented, disassortative network, to
a soft-hierarchical, peering-oriented, assortative network. This change is indicative of evolving business
relationships amongst organisations
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